Question

What are the roots of the polynomial function p(x)=x3+4x2−15x−18
?(1 point)
Responses

x=3
, x=1
and x=6
x equals 3 , x equals 1 and x equals 6

x=−3
, x=−1
and x=6
x equals negative 3 , x equals negative 1 and x equals 6

x=−6
, x=−1
and x=3
x equals negative 6 , x equals negative 1 and x equals 3

x=−6
, x=1
and x=3

Answer 1

To find the roots of the polynomial \( p(x) = x^3 + 4x^2 - 15x - 18 \), we can use the Rational Root Theorem to test possible rational roots, or we can try synthetic division with various integer values.

Let's test some integer roots:

1. **Testing \( x = -3 \)**:
\[
p(-3) = (-3)^3 + 4(-3)^2 - 15(-3) - 18 = -27 + 36 + 45 - 18 = 36 \quad \text{(not a root)}
\]

2. **Testing \( x = -1 \)**:
\[
p(-1) = (-1)^3 + 4(-1)^2 - 15(-1) - 18 = -1 + 4 + 15 - 18 = 0 \quad \text{(root)}
\]

3. **Now that we found \( x = -1 \) is a root**, we can divide the polynomial by \( x + 1 \) using synthetic division.

Perform synthetic division with \( -1 \):
```
-1 | 1 4 -15 -18
| -1 -3 18
----------------------
1 3 -18 0
```
This gives us \( x^2 + 3x - 18 \).

4. **Next, we can factor \( x^2 + 3x - 18 \)**. To factor it, we look for two numbers that multiply to -18 and add to 3. The numbers 6 and -3 work:
\[
x^2 + 3x - 18 = (x + 6)(x - 3)
\]

5. Therefore, the polynomial can be expressed as:
\[
p(x) = (x + 1)(x + 6)(x - 3)
\]

The roots of the polynomial are:
- \( x = -1 \)
- \( x = -6 \)
- \( x = 3 \)

Thus, the complete list of roots is:
- \( x = -6 \)
- \( x = -1 \)
- \( x = 3 \)

So, the correct answer is:
**x equals negative 6, x equals negative 1, and x equals 3.**